Pointwise estimates and existence of solutions of porous medium and $p$-Laplace evolution equations with absorption and measure data

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Pointwise estimates and existence of solutions of porous medium and p-Laplace evolution equations with

absorption and measure data

Marie-Françoise Bidaut-Véron, Quoc-Hung Nguyen

To cite this version:

Marie-Françoise Bidaut-Véron, Quoc-Hung Nguyen. Pointwise estimates and existence of solutions of porous medium andp-Laplace evolution equations with absorption and measure data. 2014. �hal- 01020999v2�

Pointwise estimates and existence of solutions of porous medium and p-Laplace evolution equations with absorption and measure

data

Marie-Fran¸coise Bidaut-V´eron^∗ Quoc-Hung Nguyen^†

Abstract

Let Ω be a bounded domain ofR^N(N ≥2). We obtain a necessary and a sufficient condition, expressed in terms of capacities, for existence of a solution to the porous medium equation with absorption







ut−∆(|u|^m−1u) +|u|^q−1u=µ in Ω×(0, T), u= 0 on∂Ω×(0, T),

u(0) =σ,

where σ and µ are bounded Radon measures, q > max(m,1), m > ^N−2_N . We also obtain a sufficient condition for existence of a solution to thep-Laplace evolution equation







ut−∆pu+|u|^q−1u=µ in Ω×(0, T), u= 0 on∂Ω×(0, T),

u(0) =σ.

where q > p−1 andp >2.

Contents

1 Introduction and main results 2

2 Porous medium equation 5

∗Laboratoire de Math´ematiques et Physique Th´eorique, Facult´e des Sciences, Universit´e Fran¸cois Rabelais, Tours, France. E-mail: [email protected]

†Laboratoire de Math´ematiques et Physique Th´eorique, Facult´e des Sciences, Universit´e Fran¸cois Rabelais, Tours, France. E-mail: [email protected]

3 p−Laplacian evolution equation 16 3.1 Distribution solutions . . . 16 3.2 Renormalized solutions . . . 16 3.3 Proof of Theorem 1.5. . . 19 keywords: Sobolev-Besov capacities; Bessel capacities; Radon measures; renormalized solutions.

MSC: 35K92; 35K55; 35K15

1 Introduction and main results

Let Ω be a bounded domain ofR^N, N ≥2 and T >0, and ΩT = Ω×(0, T). In this paper we study the existence of solutions to the following two types of evolution problems: the porous medium problem with absorption







ut−∆(|u|^m−1u) +|u|^q−1u=µ in ΩT, u= 0 on∂Ω×(0, T),

u(0) =σ,

(1.1) wherem > ^N−2_N andq >max(1, m),and thep-Laplace evolution problem with absorption







ut−∆pu+|u|^q−1u=µ in ΩT, u= 0 on∂Ω×(0, T),

u(0) =σ,

(1.2) whereq > p−1 >1,and µand σare bounded Radon measures respectively on ΩT and Ω. In the sequel, for any bounded domainO of R^l(l ≥1), we denote by Mb(O) the set of bounded Radon measures in O, and byM⁺_b(O) its positive cone. For anyν ∈ Mb(O),we denote byν⁺andν⁻ respectively its positive and negative part.

Whenm= 1, p= 2 andq >1 the problem has been studied by Brezis and Friedman [12] withµ= 0.It is shown that in the subcritical caseq <1 + 2/N, the problem can be solved for anyσ∈ Mb(Ω),and it has no solution whenq≥1 + 2/N andσis a Dirac mass. The general case has been solved by Baras and Pierre [5] and their results are expressed in terms of capacities. Fors >1, α >0, the capacity CapG_α,s of a Borel setE⊂R^N, defined by

CapG_α,s(E) = inf{||g||^s_Ls(R^N):g∈L^s₊(R^N),G_α∗g≥1 onE},

whereG_αis the Bessel kernel of orderαand the capacity Cap_2,1,sof a compact setK⊂R^N+1is defined by Cap_2,1,s(K) = infn

||ϕ||^s_W2,1

s (R^N+1):ϕ∈S(R^N+1), ϕ≥1 in a neighborhood of Ko , where

||ϕ||_Ws^2,1_(RN+1)=||ϕ||_L^s_(R^N+1₎+||ϕt||_L^s_(R^N+1₎+|| |∇ϕ| ||_L^s_(R^N+1₎+ X

i,j=1,2,...,N

||ϕxixj||_L^s_(R^N+1₎. The capacity Cap_2,1,s is extended to Borel sets by the usual method. Note the relation between the two capacities:

C⁻¹CapG

2−2,s(E)≤Cap_2,1,s(E× {0})≤CCapG

2−2,s(E)

for any Borel set E ⊂R^N, see [34, Corollary 4.21]. In particular, for any ω ∈ Mb(R^N) and a ∈ R, the measureω⊗δ{t=a}inR^N+1is absolutely continuous with respect to the capacity Cap_2,1,s( inR^N+1) if and only ifω is absolutely continuous with respect to the capacity CapG

2−2

s,s (inR^N).

From [5], the problem







ut−∆u+|u|^q−1u=µ in ΩT, u= 0 on∂Ω×(0, T), u(0) =σ,

has a solution if and only if the measuresµandσ are absolutely continuous with respect to the capacities Cap_2,1,q^′ in ΩT and CapG₂

q,q^′ in Ω respectively, whereq^′ =_q−1^q . In Section 2 we study problem (1.1).

For m > 1, Chasseigne [14] has extended the results of [12] for µ = 0 in the new subcritical range m < q < m+_N². The supercritical caseq≥m+_N² withµ= 0 andσis positive is studied in [13]. He has essentially proved that if problem (1.1) has a solution, thenσ⊗δ{t=0}is absolutely continuous with respect to the capacity Cap_2,1, ^q

q−m,q^′, defined for anycompact setK⊂R^N+1 by Cap_2,1, ^q

q−m,q^′(K) = inf (

||ϕ||

q q−m

W^2,1q

q−m,q′(R^N+1):ϕ∈S(R^N), ϕ≥1 in a neighborhood of E )

, where

||ϕ||_W^2,1_q

q−m,q′(R^N+1)=||ϕ||

L

q

q−m(R^N+1)+||ϕt||_Lq′

(R^N+1)+|| |∇ϕ| ||

L

q

q−m(R^N+1)+ X

i,j=1,2,...,N

||ϕxixj||

L

q

q−m(R^N+1).

In this Section, we first givenecessary conditions on the measuresµandσfor existence, which cover the results mentioned above.

Theorem 1.1 Let q > max(1, m) andµ ∈ Mb(ΩT) and σ∈ Mb(Ω). If problem (1.1) has a very weak solution thenµ andσ⊗δ{t=0} are absolutely continuous with respect to the capacity Cap_2,1, ^q

q−m,_q−1^q . Remark 1.2 It is easy to see that the capacity Cap_2,1, ^q

q−m,_q−1^q is absolutely continuous with respect to the capacity Cap_2,1, ^q

q−max{m,1}. Thereforeµandσ⊗δ{t=0} are absolutely continuous with respect to the capacities Cap_2,1, ^q

q−max{m,1}.In particularσis absolutely continuous with respect to the capacity CapG_{2 max{m,1}}

q

,_q−max{m,1}^q .

The main result of this Section is the followingsufficient condition for existence, where we use the notion ofR-truncated Riesz parabolic potentialI₂onR^N+1 of a measureµ∈ M⁺_b(ΩT) , defined by

I^R

2[µ](x, t) = Z R

0

µ( ˜Qρ(x, t)) ρ^N

dρ

ρ for any (x, t)∈R^N+1, withR∈(0,∞], and ˜Qρ(x, t) =Bρ(x)×(t−ρ², t+ρ²).

Theorem 1.3 Letm > ^N_N⁻²,q >max(1, m),µ∈ Mb(ΩT)andσ∈ Mb(Ω).

i. Ifm >1andµandσare absolutely continuous with respect to the capacities Cap_2,1,q^′ inΩT and CapG₂

q

,q^′

in Ω,then there exists a very weak solution uof (1.1), satisfying for a.e.(x, t)∈ΩT

|u(x, t)| ≤C

|σ|(Ω) +|µ|(ΩT) d^N

m1

+|σ|(Ω) +|µ|(ΩT) + 1 +I^2d₂ [|σ| ⊗δ{t=0}+|µ|](x, t)

, (1.3) where C=C(N, m)>0 and

m1= (N+ 2)(2mN+ 1)

m(mN+ 2)(1 + 2N), d=diam(Ω) +T^1/2.

ii. If ^N_N⁻² < m≤1,andµandσare absolutely continuous with respect to the capacities Cap_2,1, ^2q

2(q−1)+N(1−m)

in ΩT and CapG_2−N(1−m)

q

,2(q−1)+N(1−m)^2q in Ω, there exists a very weak solution u of (1.1), such that for a.e.(x, t)∈ΩT

|u(x, t)| ≤C

|σ|(Ω) +|µ|(ΩT) d^N

m2

+ 1 + I^2d₂ [|σ| ⊗δ{t=0}+|µ|](x, t)2−N(1−m)²

, (1.4)

where C=C(N, m)>0 and

m2= 2N(N+ 2)(m+ 1)

(2 +N m)(2−N(1−m))(2 +N(1 +m)). .

Remark 1.4 These estimates are not homogeneous in u. In particular if µ ≡ 0, u satisfies the decay estimates, fora.e.(x, t)∈ΩT,

i. if m >1,

|u(x, t)| ≤C

|σ|(Ω) d^N

m1

+|σ|(Ω) + 1 + |σ|(Ω) N t^N/2

,

ii. ifm <1,

|u(x, t)| ≤C

|σ|(Ω) d^N

m2

+ 1 +

|σ|(Ω) N t^N/2

_2−N(m−1)² ! .

We also give other types ofsufficient conditions for measures which are good in time, that means such that

σ∈L¹(Ω) and|µ| ≤f +ω⊗F, wheref ∈L¹₊(ΩT), F ∈L¹₊((0, T)), (1.5) see Theorem 2.10. The proof is based on estimates for the stationary problem in terms of elliptic Riesz potential.

In Section 3, we consider problem (1.2). Let us recall some former results about it.

Forq > p−1>0,Pettitta, Ponce and Porretta [36] have proved that it admits a (unique renormalized) solution providedσ∈L¹(Ω) andµ∈ Mb(ΩT) is adiffuse measure, i.e. absolutely continuous with respect toCp-capacity in ΩT, defined on a compact setK⊂ΩT by

Cp(K,ΩT) = inf{||ϕ||W :ϕ∈C_c^∞(ΩT)ϕ≥1 onK}, (1.6) where

W ={z:z∈L^p(0, T, W₀^1,p(Ω)∩L²(Ω)), zt∈L^p′(0, T, W^−1,p′(Ω) +L²(Ω))}.

In the recent work [7], we have proved a stability result for thep-Laplace parabolic equation, see Theorem 3.5, forp > ^2N+1_N+1. As a first consequence, in the new subcritical range

q < p−1 + p N,

problem (1.2) admits a renormalized solution for any measures µ ∈ Mb(ΩT) andσ ∈ L¹(Ω). Moreover, we have obtained sufficient conditions for existence, for measures that have agood behavior in time, of the form (1.5). It is shown that (1.2) has a renormalized solution ifω ∈ M⁺_b(Ω) is absolutely continuous with respect to CapG_p,_q−p+1^q . The proof is based on estimates of [8] for the stationary problem which involve Wolff potentials.

Here we givenew sufficient conditions when p >2.The next Theorem is our second main result:

Theorem 1.5 Let q > p−1>1 and µ∈ Mb(ΩT)andσ∈ Mb(Ω). If µandσ are absolutely continuous with respect to the capacities Cap_2,1,q′ inΩT and CapG₂

q

,q^′ inΩ, then there exists a distribution solution of problem (1.2) which satisfies the pointwise estimate

|u(x, t)| ≤C

1 +D+

|σ|(Ω) +|µ|(ΩT) D^N

m3

+I^2D₂

|σ| ⊗δ{t=0}+|µ|

(x, t)

(1.7) for a.e(x, t)∈ΩT withC=C(N, p)and

m3= (N+p)(λ+ 1)(p−1)

((p−1)N+p)(1 +λ(p−1)), λ= min{1/(p−1),1/N}, D=diam(Ω) +T^1/p. (1.8) Moreover, ifσ∈L¹(Ω),uis a renormalized solution.

2 Porous medium equation

Fork >0 and s∈R we setTk(s) = max{min{s, k},−k}. The solutions of (1.1) are considered in a weak sense:

Definition 2.1 Let µ∈ Mb(ΩT)andσ∈ Mb(Ω) andg∈C(R).

i. A function uis a weak solution of problem







ut−∆(|u|^m−1u) +g(u) =µ inΩT, u= 0 on∂Ω×(0, T),

u(0) =σ in Ω.

(2.1)

ifu∈C([0, T] ;L²(Ω)),|u|^m∈L²((0, T);H₀¹(Ω)) andg(u)∈L¹(ΩT), and for any ϕ∈C_c^2,1(Ω×[0, T)),

− Z

ΩT

uϕtdxdt+ Z

ΩT

∇(|u|^m−1u).∇ϕdxdt+ Z

ΩT

g(u)ϕdxdt= Z

ΩT

ϕdµ+ Z

Ω

ϕ(0)dσ.

ii. A functionuis a very weak solution of (2.1) if u∈L^max{m,1}(ΩT)and g(u)∈L¹(ΩT),and for any ϕ∈C_c^2,1(Ω×[0, T)),

− Z

ΩT

uϕtdxdt− Z

ΩT

|u|^m−1u∆ϕdxdt+ Z

ΩT

g(u)ϕdxdt= Z

ΩT

ϕdµ+ Z

Ω

ϕ(0)dσ.

First we give a priori estimates for the problem without perturbation term:

Proposition 2.2 Letu∈L^∞(ΩT)with |u|^m∈L²((0, T);H₀¹(Ω)) be a weak solution to problem







ut−∆(|u|^m−1u) =µ inΩT, u= 0 on∂Ω×(0, T), u(0) =σ in Ω,

(2.2)

withσ∈Cb(Ω) andµ∈Cb(ΩT). Then,

||u||L^∞((0,T);L¹(Ω))≤ |σ|(Ω) +|µ|(ΩT), (2.3)

||u||_L^m+2/N,∞_(ΩT)≤C1(|σ|(Ω) +|µ|(ΩT))^mN+2N+2 , (2.4)

|||∇(|u|^m−1u)|||

L

mN+2 mN+1,∞

(ΩT)≤C2(|σ|(Ω) +|µ|(ΩT))^m(N+1)+1mN+2 , (2.5) whereC1=C1(N, m), C2=C2(N, m).

Proof of Proposition 2.2. For anyτ∈(0, T),andk >0 we have Z

Ωτ

(Hk(u))tdxdt+ Z

Ωτ

|∇Tk(|u|^m−1u)|²dxdt= Z

Ωτ

Tk(|u|^m−1u)dµ(x, t), whereH(a) =Ra

0 Tk(|y|^m−1y)dy. This leads to Z

ΩT

|∇Tk(|u|^m−1u)|²dxdt≤k(|σ|(Ω) +|µ|(ΩT)) and (2.6) Z

Ω

(Hk(u))(τ)dx≤k(|σ|(Ω) +|µ|(ΩT)), ∀τ∈(0, T).

SinceHk(a)≥k(|a| −k) for anyaandk >0, we find Z

Ω

(|u|(τ)−k)dx≤ |σ|(Ω) +|µ|(ΩT), ∀τ∈(0, T).

Lettingk→0, we get (2.3).

Next we prove (2.4). By the Gagliardo-Nirenberg embedding theorem, there holds Z

ΩT

|Tk(|u|^m−1u)|^2(N+1)N dxdt≤C1||Tk(|u|^m−1u)||^2/N_L∞((0,T);L¹(Ω))

Z

ΩT

|∇Tk(|u|^m−1u)|²dxdt

≤C1k^2(m−1)mN ||u||^2/N_L∞((0,T);L¹(Ω))

Z

ΩT

|∇Tk(|u|^m−1u)|²dxdt.

Thus, from (2.6) and (2.3) we get k^2(N+1)N |{|u|^m> k}| ≤

Z

ΩT

|Tk(|u|^m−1u)|^2(N+1)N dxdt≤c1k^{2(m−1)mN +1}(|σ|(Ω) +|µ|(ΩT))^N+2N , which implies (2.4). Finally, we prove (2.5). Thanks to (2.6) and (2.4) we have fork, k0>0

|{|∇(|u|^m−1u)|> k}| ≤ 1 k²

Z k² 0

|{|∇(|u|^m−1u)|> ℓ}|dℓ

≤ |{|u|^m> k0}|+ 1 k²

Z

ΩT

|∇Tk0(|u|^m−1u)|²dxdt

≤C1k⁻₀ ^{mN2 −1}(|σ|(Ω) +|µ|(ΩT))^N+2N +k0k⁻²(|σ|(Ω) +|µ|(ΩT)).

Choosingk0=k^{N m+1N m} (|σ|(Ω) +|µ|(ΩT))^{N m+1m} ,we get (2.5).

Next we show the necessary conditions given at Theorem 1.1.

Proof of Theorem 1.1. As in [5, Proof of Proposition 3.1], it is enough to claim that for any compact K⊂Ω×[0, T) such that µ⁻(K) = 0, (σ⁻⊗δ{t=0})(K) = 0 and Cap_2,1, ^q

q−m,q^′(K) = 0 thenµ⁺(K) = 0 and (σ⁺⊗δ{t=0})(K) = 0. Let ε >0 and choose an open set O such that (|µ|+|σ| ⊗δ{t=0})(O\K) < εand K⊂O⊂Ω×(−T, T). One can find a sequence{ϕn} ⊂C_c^∞(O) which satisfies 0≤ϕn≤1, ϕn|_K = 1 and ϕn →0 inW^2,1q

q−m,q^′(R^N+1) and almost everywhere inO (see [5, Proposition 2.2]). We get Z

ΩT

ϕndµ+ Z

Ω

ϕn(0)dσ=− Z

ΩT

u(ϕn)tdxdt− Z

ΩT

|u|^m−1u∆ϕndxdt+ Z

ΩT

|u|^q−1uϕndxdt

≤(||u||L^q(ΩT)+||u||^m_Lq(ΩT))||ϕn||_W^2,1_q

q−m, q q−1

(R^N+1)+ Z

ΩT

|u|^qϕndxdt.

Note that Z

ΩT

ϕndµ+ Z

Ω

ϕn(0)dσ≥µ⁺(K) + (σ⁺⊗δ_{t=0})(K)−(|µ|+|σ| ⊗δ_{t=0})(O\K)

≥µ⁺(K) + (σ⁺⊗δ{t=0})(K)−ε.

This implies

µ⁺(K) + (σ⁺⊗δ{t=0})(K)≤(||u||L^q(ΩT)+||u||^m_L^q_(ΩT))||ϕn||_W^2,1_q

q−m, q q−1

(R^N+1)+ Z

ΩT

|u|^qϕndxdt+ε.

Letting the limit we getµ⁺(K) + (σ⁺⊗δ{t=0})(K)≤ε. Therefore,µ⁺(K) = (σ⁺⊗δ{t=0})(K) = 0.

Next we look for sufficient conditions of existence. The crucial result used to establish Theorem 1.3 is the following a priori estimates, due to of Liskevich and Skrypnik [31] form≥1 and Bogelein, Duzaar and Gianazza [11] form≤1.

Theorem 2.3 Letm > ^N_N⁻² andµ∈(Cb(ΩT))⁺. Letu∈L^∞₊(ΩT)withu^m∈L²(0, T, H_loc¹ (Ω))be a weak solution to equation

ut−∆(u^m) =µ inΩT.

Then there existsC=C(N, m)such that, for almost all(y, τ)∈ΩT and any cylinderQ˜r(y, τ)⊂⊂ΩT,there holds

i. if m >1

u(y, τ)≤C



 1 r^N+2

Z

Q˜r(y,τ)

|u|^m+2N1 dxdt

!_1+2N^2N

+||u||L^∞((τ−r²,τ+r²);L¹(Br(y)))+ 1 +I^2r₂ [µ](y, τ)



,

ii. ifm≤1,

u(y, τ)≤C



 1 r^N+2

Z

Q˜r(y,s)

|u|^{2(1+mN)N(1+m)}dxdt

!(2−N(1−m))(2+N(1+m))^2N(m+1)

+ 1 + I^2r

2 [µ](y, τ)_2−N(1−m)²



.

As a consequence we get a new a priori estimate for the porous medium equation:

Corollary 2.4 Letm > ^N−2_N andµ∈Cb(ΩT). Let u∈L^∞(ΩT) with|u|^m∈L²(0, T, H₀¹(Ω)) be the weak solution of problem







ut−∆(|u|^m−1u) =µ inΩT, u= 0 on∂Ω×(0, T), u(0) = 0 in Ω.

Then there existsC=C(N, m)such that, for a.e. (y, τ)∈ΩT, i. if m >1,

|u(y, τ)| ≤C

|µ|(ΩT) d^N

m1

+|µ|(ΩT) + 1 +I^2d₂ [|µ|](y, τ)

, (2.7)

ii. ifm≤1,

|u(y, τ)| ≤C

|µ|(ΩT) d^N

m2

+ 1 +

I^2d₂ ¹[|µ|](y, τ)_2−N(1−m)²

, (2.8)

wherem1, m2 anddare defined in Theorem 1.3.

Proof. Let x0 ∈ Ω, and Q = B2d(x0)×(−(2d)²,(2d)²). Consider the function U ∈ (Cb(Q))⁺, with U^m∈L^p((−(2d)²,(2d)²);H₀¹(B2d(x0))) such that U is weak solution of







Ut−∆(U^m) =χΩT|µ| inB2d(x0)×(−(2d)²,(2d)²), U = 0 on∂B2d(x0)×(−(2d)²,(2d)²), U(−(2d)²) = 0 in B2d(x0).

(2.9) From Theorem 2.3, we get, for a.e (y, τ)∈ΩT,

U(y, τ)≤c1



 1 d^N+2

Z

Q˜d(y,τ)

|U|^m+2N1 dxdt

!_1+2N^2N

+||U||L^∞((τ−d²,τ+d²);L¹(Bd(y)))+ 1 +I^2d

2 [|µ|](y, τ)





ifm >1 and U(y, τ)≤C



 1 d^N+2

Z

Q˜d(y,s)

|u|^{2(1+mN)N(1+m)}dxdt

!(2−N(1−m))(2+N(1+m))^2N(m+1)

+ 1 + I^2r₂ [µ](y, τ)_2−N(1−m)²





ifm≤1. By Proposition 2.2, we have

||U||L^∞((τ−d²,τ+d²);L¹(Bd(y)))≤ |µ|(ΩT),

|{|U|> ℓ}| ≤c2(|µ|(ΩT))^2+NN ℓ^−N2−m, ∀ℓ >0.

Thus, for anyℓ0>0, Z

Q

U^m+2N1 dxdt= (m+ 1 2N)

Z ∞ 0

ℓ^{m+2N1 −1}|{U > ℓ}|dℓ

= (m+ 1 2N)

Z ℓ0

0

ℓ^{m+2N1 −1}|{U > ℓ}|dℓ+ (m+ 1 2N)

Z ∞ ℓ0

ℓ^{m+2N1 −1}|{U > ℓ}|dℓ

≤c3d^N+2ℓ^m+₀ ^2N1 +c4ℓ₀^{2N1 −N2}(|µ|(ΩT))^2+NN . Choosingℓ0=_|µ|(Ω

T) d^N

_mN^N+2₊₂

, we get Z

Q

U^{(λ+1)(p−1)}dxdt≤c5d^N+2

|µ|(ΩT) d^N

(N+2)(2mN+1) 2mN(mN+2)

. Thus, for a.e (y, τ)∈ΩT,

U(y, τ)≤c6

|µ|(ΩT) d^N

m1

+|µ|(ΩT) + 1 +I^2d₂ [|µ|](y, τ)

ifm >1. Similarly, we also obtain for a.e (y, τ)∈ΩT, U(y, τ)≤c7

|µ|(ΩT) d^N

m2

+ 1 +

I^2d₂ ¹[|µ|](y, τ)_2−N(1−m)² . ifm≤1. By the comparison principle we get|u| ≤U in ΩT, and (2.7)-(2.8) follow.

Lemma 2.5 Letg∈Cb(R)be nondecreasing withg(0) = 0, andµ∈Cb(ΩT). There exists a weak solution u∈L^∞(ΩT)with |u|^m∈L²(0, T, H₀¹(Ω)) of problem







ut−∆(|u|^m−1u) +g(u) =µ inΩT, u= 0 on∂Ω×(0, T),

u(0) = 0 in Ω.

(2.10)

Moreover, the comparison principle holds for these solutions: ifu1, u2 are weak solutions of (2.10) when (µ, g) is replaced by (µ1, g1) and (µ2, g2), where µ1, µ2 ∈ Cb(ΩT) with µ1 ≥ µ2 and g1, g2 have the same properties asg with g1≤g2 inRthen u1≥u2 in ΩT.

As a consequence, ifµ≥0 then u≥0.

Proof of Lemma 2.5. Set an(s) = m|s|^m−1 if 1/n≤ |s| ≤ n and an(s) = m|n|^m−1 if |s| ≥n, an(s) =m(1/n)^m−1if|s| ≤1/n. AlsoAn(τ) =Rτ

0 an(s)ds. Then one can findun being a weak solution to the following equation







(un)t−div(an(un)∇un) +g(un) =µ in ΩT, un= 0 on∂Ω×(0, T),

un(0) = 0 in Ω.

(2.11) It is easy to see that|un(x, t)| ≤t||µ||L^∞(ΩT) for all (x, t)∈ΩT. Thus, choosingAn(un) as a test function,

we obtain Z

ΩT

|∇An(un)|²dxdt≤C1(T,||µ||L^∞(ΩT)). (2.12) Now set Φn(τ) = Rτ

0 |An(s)|ds. Choosing|An(un)|ϕ as a test function in (2.11), whereϕ∈ C_c^2,1(ΩT), we get the relation in D^′(ΩT) :

(Φn(un))t−div(|An(un)|∇An(un)) +∇An(un).∇|An(un)|+|An(un)|g(un) =|An(un)|µ.

Hence,

||(Φn(un))t||_L¹_(ΩT)+L²_((0,T);H⁻¹_(Ω)) ≤ ||An(un)∇An(un)||L²(ΩT)+||∇An(un)|||²_L2(ΩT)

+||An(un)g(un)||L¹(ΩT)+||An(un)µ||L¹(ΩT). Combining this with (2.12) and the estimate|An(un)| ≤C2(T,||µ||L^∞(Ω)), we deduce that

sup

n

||(Φn(un))t||_L¹_(ΩT)+L²_(0,T,H⁻¹_(Ω)) <∞.